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Full text of "Analytical Mechanics"

ANGULAR IMPULSE AND ANGULAR MOMENTUM    269
remains constant, in direction as well as in magnitude.    This is the principle of the conservation of angular momentum.
ILLUSTRATIVE EXAMPLE.
Discuss the effect of a shrinkage in the radius of the earth upon the length of the day.
Let P and Pr be the lengths of the day when the radius of the earth is a and a', respectively. Further, let co and a/ be the corresponding values of the angular velocity of the earth about its axis. Then
But since the earth is not supposed to be acted upon by any external torques its angular momentum remains constant. Therefore
Ice - IV.                                          (2)
From equations (1) and (2) we obtain
H' - L _ 2!!
P ~~ I ~ a2'
p - p'         q2 _ a/2
or                                p -
,                                          5P     a+a'  Sa                                  ,QX
and                                               -^- = -------->                                    (6)
P          a      a
where dP and da denote the diminutions in the length of the day and the radius, respectively. When da is small a' is very nearly equal to a, therefore equation (3) may be written in-the form
Therefore the percentage diminution in the length of the day is twice as large as the percentage diminution in the radius. Hence when the radius is diminished by 1 mile the length of the day is diminished by about 43 seconds.
PROBLEMS.
1.   How do the oceanic currents from the polar regions affect the length of the day?
2.   A uniform rod of negligible diameter falls from a vertical position with its lower end on a perfectly smooth horizontal plane.   What is the path of its middle point?